M15 - Time Series Analysis Flashcards
Time series analysis
- … unit if observation
- …. points in time
Describes the …. change in y
Used for ….
- 1 unit if observation
- various points in time
Describes the temporal change in y
Used for forecasting
Portfolio mgmt:
…-French …-factor model
“Observation that … …. of … have tended to be … than the market
–> (I) small … (II) stocks with … price-to …. ratio
CAPM : capital … … mgmt
“Uses … variable to describe the … of a portfolio”
–> add those … factors to capm to reflect the portfolio’s …. to these ….
Portfolio mgmt:
FAMA-French 3-factor model
“Observation that 2 CLASSES of STOCKS have tended to be BETTER than the market
–> (I) small CAPS (II) stocks with LOW price-to BOOK ratio
CAPM : capital ASSET PRICING mgmt
“Uses ONE variable to describe the RETURNS of a portfolio”
–> add those 2 factors to capm to reflect the portfolio’s EXPOSURE to these FACTORS
quantitative forecasting models are based on ..
two methological subgroups:
…on a mathematical model
one or multiple time series
one series: time series extrapolation
one/ multiple: causal forecasting methods
Time series extrapolation
- classes of importance:
- These 3 classes … … on … data
- possible combinations
- autoregressive (AR) models
- integrated (I) models
- moving average (MA) models
- These 3 classes DEPENDING LINEARLY on PREVIOUS data
- autoregressive moving avergage (ARMA)
- autoregressive integrated moving average (ARIMA)
When are the time series models suitable?
suitable, if Xt can be modelled as a linear function of earlier values of Xt-1
time series extrapolation:
Gaussian White Noise
–> samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance;
at any point in time its totally random what you observe
time series extrapolation: Moving Average (MA)
earlier effect of Xt-1 and the error term still has an effect on Xt
–> the output variable depends linearly on the current and various past values of a stochastic term
time series extrapolation:
Random walk
zero of Xt is the value of Xt-1 plus error term
–> describes a path that consists of a succession of random steps
time series extrapolation:
Autoregressive
earlier point Xt-1 has an effect on Xt but to a reduced extent (beta)
–> the output variable depends linearly on its own previous values and on a stochastic term
Causal forecasting methods
- represents
- based on
- models
- a model is specified that represents the causal relationships between the variables
- based on time series data
single equation models or simultaneous models
Formulation of a model
- basic idea:
- additive time series equation:
- multiplikative time series equation:
- what are the components?
- basic idea: splitting the time series components into different components –> time series analysis decomposition
- Y = A + K + S + u
- Y = AKS*u
Y = variable to be forecasted;
systematic: A = trend component (long-term development of y), K = cyclical component, S = seasonal component (cyclical variations of y around a long-term trend
Whats the Durbin-watson statistics?
- expected values
is there autocorrelation in the residuals (prediction errors)?
- the expected value of d is large for large T’s,
…for perfect positive correlation: 0
…for complete uncorrelated terms: 2
…for perfect negative correlation: 4
T + k
T
k
T + k = forecast value for the period
T = end of observation period
k = number of observation points in the future
Name different time series models
- linear time series model
- non-linear time series models:
- -> square-root model
- -> logarithmic model
- -> multiplicative model
- -> power regression model
on what does forecasting depend?
depends on the quality, which depends on how well the model fits the data
Whats the projection interval?
With 95% confidence (alpha=0.05) it is expected that YT+k will be between value 1 and value 2 in the period T+k
Forecasting accuracy
- problem?
- solution 1
- solution 2
- the standard quality measures of regression (R², F, etc.) only tell us how well the model fits the OBSERVED values
- compare predicted values with realized ones –> but only possible after realization
- ex-post-forecast: forecast that is run in past periods but only for those periods that have available data; the last observed time series values are compared with the ex-post forecast values
Structural breaks
- def
- leads to
- how to deal with them?
- e.g.
- are unexpected shifts in time series analysis
- leads to forecasting errors and unreliablity in general
- by dummy variables: they equal zero before ths reak, afterwards they are 1
- law changes, brexit, Trump, EU expansion
Cyclical variations
- def
- how to deal with them?
- why cant we create a dummy variable for all possible states? (winter, fall, spring, summer)
- e.g.
variations in the data not due to fixed periods
last on avg. more than 1/2 years
- dummy variables
- because we would have perfect collinearity (only valid if we have alpha; if alpha wouldnt exist, we could involve all dummies)
- -> always have 1 dummy less than you have cases
difference of forecasting accuracy with cross-sectional data?
–> in cross-sectional data you have observed values, in forecasting you only have predicted ones:
the standard quality measures are not accurately
Change in trend
there is also an unexpected shift, but due to a change in the relationship
- change of interest rates
